Extreme points, support points and the Loewner variation in several complex variables

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• 作者：Ian Graham (1) graham@math.toronto.edu
  Hidetaka Hamada (2) h.hamada@ip.kyusan-u.ac.jp
  Gabriela Kohr (3) gkohr@math.ubbcluj.ro
  Mirela Kohr (3) mkohr@math.ubbcluj.ro
• 关键词：biholomorphic mapping – Loewner chain – parametric representation – reachable set – starlike mapping
• 刊名：SCIENCE CHINA Mathematics
• 出版年：2012
• 出版时间：July 2012
• 年：2012
• 卷：55
• 期：7
• 页码：1353-1366
• 全文大小：278.0 KB
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• 作者单位：1. Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada2. Faculty of Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome, Higashi-ku Fukuoka, 8138503 Japan3. Faculty of Mathematics and Computer Science, Babe?-Bolyai University, 1 M. Kog?lniceanu Str., Cluj-Napoca, 400084 Romania
• ISSN：1869-1862

文摘

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball B n in ? n . We consider the class S 0(B n ) of biholomorphic mappings on B n which have parametric representation, i.e., they are the initial elements f(·, 0) of a Loewner chain f(z, t) = e t z + … such that {e?t f(·, t)} _t?0 is a normal family on B n . We show that if f(·, 0) is an extreme point (respectively a support point) of S 0(B n ), then e ?t f(·, t) is an extreme point of S 0(B n ) for t ? 0 (respectively a support point of S 0(B n ) for t ∈ [0,t ₀] and some t ₀ > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S 0(B n ) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of B n generated by using extension operators that preserve Loewner chains.